(All images created with MATLAB® software)
Figure 2: Imaginary part of the cube root on the complex plane
Imaginary part of the cubic root function on the complex plane. Notice how the three levels of the surface join at the branch point at the origin. Notice also that the other crossings of levels occur in different places for the real and the imaginary parts. Thus, for the full surface of values of the function (i.e.: the surface in four dimensions given by the mapping from z to z1/3), the only point where the levels touch is the origin.
Picture obtained using the script PlotCube in the Athena 18.04 MATLAB® Toolkit.
Figure 2: Imaginary part of the cube root on the complex plane